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Kumar, you wrote:
"How will a free gyroscope move when initially set horizontal at the equator 
with a azimuth of say 045 degs (Not East-West)"

You can think about this in two ways. The easiest if you're just starting out 
is to think in terms of an inertial frame of reference (fixed relative to the 
distant stars). Then you can us the idea that you mentioned of a "gyro star". 
The gyroscope, free of all other forces, will continue to point towards 
whatever "gyro star" it is initially aimed at while the Earth turns 
underneath it. So in your case, the gyroscope will track a point in the sky 
45 degrees away from the celestial pole. The real star Deneb (alpha Cygni) 
has a declination of just about 45 degrees, so if you set a (completely) free 
gyroscope pointing towards Deneb it would just about match your conditions 
and the motion of Deneb in the sky would match the motion of the axis of your 
gyroscope. If you're on the equator, that means that the axis would climb 
first vertically from the horizon and then arc over towards the meridian. The 
axis would cross the meridian travelling horizontally at the an altitude of 
45 degrees six hours later. It would then descend towards the western horizon 
which it would reach six hours later, cross the lower branch of the meridian 
six hours after that and finally return to its original direction pointing NE 
and horizontal six hours after that (24 hours after you started).

When you've had more experience with these things, and especially when you 
need to deal with other forces acting on the gyroscope, then it becomes more 
useful to work in a frame of reference which is rotating with the Earth. In 
this frame of reference, there is a Coriolis acceleration which causes 
precession in any gyroscope about an axis parallel with the Earth's axis. 
It's two different ways of looking at the same thing. 

-FER




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